Construction of some algebras associated to directed graphs and related to factorizations of noncommutative polynomials

Geometry of free loci and factorization of noncommutative polynomials

The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.

Noncommutative polynomial maps

Polynomial maps attached to polynomials of an Ore extension are naturally defi ned. In this setting we show the importance of pseudo-linear transformations and give some applications. In particular, factorizations of polynomials in an Ore extension over a fi nite fi eld F_q[t;S ], where S is the Frobenius automorphism, are translated into factorizations in the usual polynomial ring F_q[x].

不可约非交换二次曲面

Irreducible noncommutative quadrics

通过非交换分次矩阵分解研究不可约非交换二次曲面，证明线模由秩2不可分解非交换线性矩阵分解的直纹描述，并证明所有光滑中心Sklyanin二次曲面都是标准的。

Algebras associated to acyclic directed graphs

We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized layered graph. We construct linear bases in such algebras and compute their Hilbert series. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of polynomials over noncommutative rings.

Factorization of noncommutative polynomials and Nullstellensätze for the free algebra

This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots,x_g)$ is $Z(f)=(Z_n(f))_n$, where $Z_n(f)=\{X \in M_n^g: \det f(X) = 0\}.$ The first main theorem of this article shows that the irreducible factors of $f$ are in a natural bijective correspondence with irreducible components of $Z_n(f)$ for every sufficiently large $n$. With each polynomial $h$ in $x$ and $x^*$ one also associates its real singularity set $Z^{re}(h)=\{X: \det h(X,X^*)=0\}$. A polynomial $f$ which depends on $x$ alone (no $x^*$ variab…

弯曲DG模与非交换二次超曲面的矩阵分解

Curved DG Modules and Matrix Factorizations from Noncommutative Quadric Hypersurfaces

本文研究非交换二次超曲面范畴的对偶性，构造从分次模到弯曲DG模同伦范畴的忠实函子，并在一定条件下将其限制到矩阵分解稳定范畴，证明偶数克利福德代数与PBW形变同构。

Noncommutative symmetric functions and W-polynomials

Let K,S,D be a division ring, an endomorphism and a S-derivation of K, respectively. In this setting we introduce generalized noncommutative symmetric functions and obtain Vieta formula and decompositions of differential operators. W-polynomials show up naturally, their connections with P-independency, Vandermonde and Wronskian matrices are briefly studied. The different linear factorizations of W-polynomials are analysed. Connections between the existence of LLCM of monic linear polynomials with coefficients in a ring and the left duo property are established at the end of the paper.

通过等变化的非交换Knörrer周期性

Noncommutative Knörrer periodicity via equivariantization

本研究利用二阶循环群作用下的等变化理论，建立了任意环上投射模分解的非交换Knörrer周期性，得到其显式拟逆，并将该周期性与大奇点范畴间的特定张量函子做了对比。

三维PI Artin-Schelter正则代数都是唯一分解环

PI Artin--Schelter regular algebras of dimension 3 are unique factorization rings

本文证明了三维PI Artin-Schelter正则代数是唯一分解环，非交换代数中对应正则局部环三维情况的类比，为Auslander-Buchsbaum定理的证明提供了关键支撑。

On a class of algebras associated to directed graphs

To any directed graph we associate an algebra with edges of the graph as generators and with relations defined by all pairs of directed paths with the same origin and terminus. Such algebras are related to factorizations of polynomials over noncommutative algebras. We also construct a basis for our algebras associated to layered graphs.

Clifford deformations of Koszul Frobenius algebras and noncommutative quadrics

Let $E$ be a Koszul Frobenius algebra. A Clifford deformation of $E$ is a finite dimensional $\mathbb Z_2$-graded algebra $E(θ)$, which corresponds to a noncommutative quadric hypersurface $E^!/(z)$, for some central regular element $z\in E^!_2$. It turns out that the bounded derived category $D^b(\text{gr}_{\mathbb Z_2}E(θ))$ is equivalent to the stable category of the maximal Cohen-Macaulay modules over $E^!/(z)$ provided that $E^!$ is noetherian. As a consequence, $E^!/(z)$ is a noncommutative isolated singularity if and only if the corresponding Clifford deformation $E(θ)$ is a semisimple …

非交换整环的不可约因子图类型，II

Types of Irreducible Divisor Graphs of Noncommutative Domains, II

本文推广非交换整环中的不可约因子图，提出不可约因子单纯复形与 τ-不可约因子图，并据此刻画 n-唯一分解整环。

The multiplicative ideal theory of Leavitt path algebras of directed graphs - a survey

Let L be the Leavitt path algebra of an arbitrary directed graph E over a field K. This survey article describes how this highly non-commutative ring L shares a number of the characterizing properties of a Dedekind domain or a Prüfer domain expressed in terms of their ideal lattices. Special types of ideals such as the prime, the primary, the irreducible and the radical ideals of L are described by means of the graphical properties of E. The existence and the uniqueness of the factorization of a non-zero ideal of L as an irredundant product of prime or primary or irreducible ideals is establis…

On the classification and properties of noncommutative duplicates

We give an explicit description of the set of all factorization structures, or twisting maps, existing between the algebras k^2 and k^2, and classify the resulting algebras up to isomorphism. In the process we relate several different approaches formerly taken to deal with this problem, filling a gap that appeared in a recent paper by Cibils. We also provide a counterexample to a result concerning the Hochschild (co)homology appeared in a paper by J.A. Guccione and J.J. Guccione.

相干$I$-标代数与非交换射影概形

Coherent $I$-indexed algebras and noncommutative projective schemes

该研究刻画了以局部有限定向偏序集$I$为指标集的相干$I$-标代数对应的非交换射影概形，拓展了Polishchuk针对相干$\boldsymbol{\text{Z}}$-标代数的相关刻画。

From quantum Ore extensions to quantum tori via noncommutative UFDs

All iterated skew polynomial extensions arising from quantized universal enveloping algebras of Kac-Moody algebras are special examples of a very large, axiomatically defined class of algebras, called CGL extensions. For the purposes of constructing initial clusters for quantum cluster algebra structures on an algebra R, and classification of the automorphisms of R, one needs embeddings of R into quantum tori T which have the property that R contains the corresponding quantum affine space algebra A. We explicitly construct such an embedding A \subseteq R \subset T for each CGL extension R usin…

Factorization in Quantum Planes

These results stem from a course on ring theory. Quantum planes are rings in two variables $x$ and $y$ such that $yx=qxy$ where $q$ is a nonzero constant. When $q=1$ a quantum plane is simply a commutative polynomial ring in two variables. Otherwise a quantum plane is a noncommutative ring. Our main interest is in quadratic forms belonging to a quantum plane. We provide necessary and sufficient conditions for quadratic forms to be irreducible. We find prime quadratic forms and consider more general polynomials. Every prime polynomial is irreducible and either central or a scalar multiple of $x…

Matrix Factorizations of the discriminant of $S_n$

Consider the symmetric group $S_n$ acting as a reflection group on the polynomial ring $k[x_1, \ldots, x_n]$, where $k$ is a field such that Char$(k)$ does not divide $n!$. We use Higher Specht polynomials to construct matrix factorizations of the discriminant of this group action: these matrix factorizations are indexed by partitions of $n$ and respect the decomposition of the coinvariant algebra into isotypical components. The maximal Cohen-Macaulay modules associated to these matrix factorizations give rise to a noncommutative resolution of the discriminant and they correspond to the nontri…

Representations of Aut(A(Gamma)) acting on homogeneous components of A(Gamma) and A(Gamma) dual

In this paper we will study the structure of algebras A(Gamma) associated to two directed, layered graphs Gamma. These are algebras associated with Hasse graphs of n-gons and the algebras Q_n related to pseudoroots of noncommutative polynomials. We will find the filtration preserving automorphism group of these algebras and then we will find the multiplicities of the irreducible representations of Aut(A(Gamma)) acting on the homogeneous components of A(Gamma) and A(Gamma) dual.

莱维特路代数的非交换零点定理

A Noncommutative Nullstellensatz for Leavitt Path Algebras

本文证明莱维特路代数的非交换零点定理，解答其商何时同构于自身的问题，定义其理想空间的莫拉不变分层与参数化，指出其与量子空间的联系并给出实例。

Dimer Models and Hochschild Cohomology

Dimer models provide a method of constructing noncommutative crepant resolutions of affine toric Gorenstein threefolds. In homological mirror symmetry, they can also be used to describe noncommutative Landau--Ginzburg models dual to punctured Riemann surfaces. For a zigzag consistent dimer embedded in a torus, we explicitly compute the Hochschild cohomology of its Jacobi algebra in terms of dimer combinatorics. This includes a full characterization of the Batalin--Vilkovisky structure induced by the Calabi--Yau structure of the Jacobi algebra. We then compute the compactly supported Hochschild…

非交换环的压缩零因子图

Compressed zero-divisor graphs of noncommutative rings

本研究将压缩零因子图概念推广至非交换环，构造保积函子，刻画有限域上矩阵环对应图的性质与自同构，证明满足条件的环同构可由其压缩零因子图同构推导。

Quantum cluster algebra structures on quantum nilpotent algebras

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to thes…

The Minimal Non-Koszul A(Gamma)

The algebras $A(Γ)$, where $Γ$ is a directed layered graph, were first constructed by I. Gelfand, S. Serconek, V. Retakh and R. Wilson. These algebras are generalizations of the algebras $Q_n$, which are related to factorizations of non-commutative polynomials. It was conjectured that these algebras were Koszul. In 2008, T.Cassidy and B.Shelton found a counterexample to this claim, a non-Koszul $A(Γ)$ corresponding to a graph $Γ$ with 18 edges and 11 vertices. We produce an example of a directed layered graph $Γ$ with 13 edges and 9 vertices which produces a non-Koszul $A(Γ)$. We also show thi…